Limits and Intro to Derivatives

Bren Calculus Workshop


Carmen Galaz García, Ph.D.

Bren School of Environmental Science & Management



Last updated: Sep 10, 2025


Materials have been adapted and expanded from Nathaniel Grimes work for the Bren Calculus Workshop.

Team Review


  • How did everyone feel about the problem set?

  • Any questions?

  • Discuss with study partner

Functions are like baking recipes


  1. Assemble all your ingredients.

In a function, these are the independent variables like \(x\).

  1. Follow the instructions to mix, bake, and decorate

These are the mathematical instructions in the function about how to manipulate the independent variables.

  1. End up with final product!

This is the ouput \(f(x)\) or dependent variable.


Typically we use the notation \(f(x)=x\), but we can always use different representations like \(g(x)=x\) or \(y=x\).

What is a function?


For each combination of independent variables, there is exactly one (and only one) value of the dependent variable.


Example

A function \(f(x)\) cannot return two values for \(x=1\): \(f(1)\ne\{2,3\}\)


Vertical line test

If you run a vertical line perpendicular to the \(x\)-axis and see where it intersects the graph of a function, then it should only intersect it once at every value of \(x\).


🤔 Are these functions?

Continuity


Intuitively, a function is continuous if there are no ‘breaks’ in the graph of the function. This means you are able to draw the whole graph without lifting your pencil.

To define it formally we need to understnad limits.


A function is discontinuous if it’s not continuous! So there are ‘breaks’ or ‘gaps’ in its graph.


Continuous

Discontinuous

Limits


Definition

We say a number \(L\) is the limit of a function \(f(x)\) as its variable \(x\) approaches a number \(c\), if the function’s output values \(f(x)\) approach \(L\) when \(x\) get closer to \(c\).


We write this symbolically as:

\[ \large \lim_{x\to c} f(x)=L \]


We read this as: “The limit of \(f(x)\) as \(x\) approaches \(c\) is \(L\).”

Example: numerical


What is the limit of \(f(x)=2x^2-4\) as \(x\) approaches 2?

We need to examine the values of \(f(x)\) as \(x\) gets closer to 2 from both sides.


From both directions it looks like \(f(x)\) converges to 4.

So, we say that “The limit of \(2x^2-4\) as \(x\) approaches 2 is 4” and we can write it like this:

\[ \lim_{x\to 2}(2x^2-4)=4 \]

Example: graph


What is the limit of \(f(x)=2x^2-4\) as \(x\) approaches 2?

From both directions it looks like \(f(x)\) converges to 4.

So, \(\lim_{x\to 2}(2x^2-4)=4\).

Continuity revisited


Finding limits of functions when f(x) is continuous at c is easy.

\[ \lim_{x\to c}f(x)=f(c) \]

What happens with discontinuous functions?


Try with your group the same approach for:

\[ \lim_{x\to 2}f(x)=\frac{|x-2|}{x-2} \]

Hint: Start very close to 2 like 1.9 and 2.1

We broke the universe


  • Dividing by zero is impossible

  • The function approaches different values from either side

  • Therefore…

Clearer on a graph


Technically the limit is “Undefined”, because it does approach a finite value. Does not exist happens when the limits spiral off to infinity in both directions (tan graph)

Defined Limit at an undefined point


  • Discontinuous functions can still have limits

  • Find the limit of \(f(x)=x+1,x\ne 2\) as x approaches 2

\(x=2\) may not exist, but we can still find a limit because it consistently approaches 3 from both directions

Team Assessment

Work with your team to complete these task

Part 1:

\[ \lim_{x\to3}g(x)=0 \]

\[ \lim_{x\to-4}g(x)=3 \]

\[ \lim_{x\to 3} g(x)= \text{DNE} \]

Work with your team to complete these task

Part 2:

  1. Can you think of examples where discontinuous functions might exist in environmental science?

  2. Choose as a team to draw an example of one of these statements:

  • \(\lim_{x\to 4^-} f(x)\) and \(\lim_{x\to 4^+}f(x)\) are both infinite

  • \(\lim_{x\to 3} f(x)=2\), but \(f(3)=0\)

  • \(\lim_{x\to 5^-} f(x)=4\) and \(\lim_{x\to 5^+} f(x)=2\)

  • \(\lim_{x\to -3} f(x)=-5\) but \(f(-3)=-5\)

Introduction to Derivatives

Putting it all together


Recall average rate of change and instantaneous

Taking the average rate of change to a set limit will eventually converge to the instantaneous.

Walk Through Example


Walk Through Example


Walk Through Example


Walk Through Example


Walk Through Example


Walkthrough


Tangent Lines


What if we set \(\Delta x=0\)? Then we would have a slope line that only touches our function at exactly \(x\).

These are called tangent lines

Put our example in math notation


\[\text{slope}=\frac{y_2-y_1}{x_2-x_1}\]

Choose a point along the function \((x,f(x))\)

Choose a different point on the function \(\Delta x\) away \((x+\Delta x,f(x+\Delta x))\)

Add these into the slope equation

\[ \text{slope}=\frac{f(x+\Delta x)-f(x)}{(x+\Delta x) -x}=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]

Derivative Definition


\[ \large f'(x)=\lim_{\Delta x \to 0}=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]


Most common notation

\[ \large f'(x)\text{, or }\frac{dy}{dx} \]

Not all functions are differentiable


  • All differentiable functions are continuous, but not all continuous functions are differentiable

  • The absolute value function is one example \(y=|x|\)

Calculus and Derivatives are the study of change


Environmental Science is also a study of change


Environmental Science is also a study of change


Environmental Science is also a study of change


Rules for Differentiation


Constant Rule

  • If \(f(x)\) is contant, then for all \(x\), \(f'(x)=0\)

\[ \begin{align} &y=a &\frac{dy}{dx}=0 \end{align} \]

Power Rule

\[ \frac{d}{dx}[x^n] =nx^{n-1} \]

Examples


\[ \begin{align} &y=100 & &y=5x^5 & &y=\frac{1}{x^2} \end{align} \]

Rules for Differentiation


Sum and Difference Rules

\[ \begin{align} \frac{d}{dx}&=[f(x)+g(x)]=f'(x)+g'(x) \\ \frac{d}{dx}&=[f(x)-g(x)]=f'(x)-g'(x) \end{align} \]

Looks really scary. All it says, if the function has pieces that are added or subtracted you can take the derivative of each individual piece.

Example


\[ \begin{align} &y=x^2+8x+4 & &y=x^3-x^2+x-15 \end{align} \]

Team Assessment

  1. As a team, list 5 fields of environmental science where studying the rate of change and derivatives would be important


  1. Which rules should you use to take these derivatives?

\[ \begin{align} \text{A) }& f(x)=3x^4 &\text{B) } y=4x^2+3x-16 \end{align} \]

  1. Find the derivatives of these functions

\[ \begin{align} &\text{A) } y=3x^2 & &\text{B) }h(x)=y=7x+4 & &\text{C) }g(y)=\sqrt{y} \end{align} \]

  1. Discuss why derivatives need continuity. It might help to think about why some continuous functions don’t have derivatives.